Related papers: Compatibility of Gauss maps with metrics

We use a new method to give conditions for the existence of a local isometric immersion of a Riemannian $n$-manifold $M$ in $\mathbb{R}^{n+k}$, for a given $n$ and $k$. These equate to the (local) existence of a $k$-tuple of scalar fields…

We introduce the notion of translational Riemannian manifolds and define a Gauss map for orientable immersed hypersurfaces lying in these ambients, an associated translational curvature and prove a Gauss-Bonnet theorem. We also use this…

In this article, we give the integrability conditions for the existence of an isometric immersion from an orientable simply connected surface having prescribed Gauss map and positive extrinsic curvature into some unimodular Lie groups. In…

Let $M$ be a connected, non-compact $m$-dimensional Riemannian manifold. In this paper we consider smooth maps $\phi: M \to \mathbb{R}^n$ with images inside a non-degenerate cone. Under quite general assumptions on $M$, we provide a lower…

The Gauss map of a conformal minimal immersion of an open Riemann surface $M$ into $\mathbb R^3$ is a meromorphic function on $M$. In this paper, we prove that the Gauss map assignment, taking a full conformal minimal immersion $M\to\mathbb…

We investigate isometric immersions $f\colon M^n\to\R^{n+2}$, $n\geq 3$, of Riemannian manifolds into Euclidean space with codimension two that admit isometric deformations that preserve the metric of the Gauss map. In precise terms, the…

We consider conformal immersions of Riemann surfaces in $\bb{S}^4$ and study their Gauss maps with values in the Grassmann bundle $\mathcal{F} = SO_5/T^2 \to \mathbb{S}^4$. The energy of maps from Riemann surfaces into $\mathcal{F}$ is…

We consider immersions of a Riemann surface into a manifold with $G_2$-holonomy and give criteria for them to be conformal and harmonic, in terms of an associated Gauss map.

In this paper we investigate the existence of ``partially'' isometric immersions. These are maps f:M->R^q which, for a given Riemannian manifold M, are isometries on some sub-bundle H of TM. The concept of free maps, which is essential in…

Let $\psi:\M \to \SH$ be an isometric immersion of codimension 1, then there exist symmetric $(1,1)$-tensors $S$ and $f$, a tangent vector field $U$ and a smooth function $\lambda$ on $\M$ that satisfy the compatibility equations of $\SH$.…

The Gauss map of a generic immersion of a smooth, oriented surface into $\mathbb R^4$ is an immersion. But this map takes values on the Grassmanian of oriented 2-planes in $\mathbb R^4$. Since this manifold has a structure of a product of…

We study generic Riemannian submersions from nearly Kaehler manifolds onto Riemannian manifolds. We investigate conditions for the integrability of various distributions arising for generic Riemannian submersions and also obtain conditions…

An isometric embedding of a graph into a metric space is an embedding of the vertices such that the smallest number of edges connecting any two vertices equals to the distance in the metric space between the images. In this paper, we study…

For an oriented isometric immersion $f:M\to S^n$ the spherical Gauss map is the Legendrian immersion of its unit normal bundle $UM^\perp$ into the unit sphere subbundle of $TS^n$, and the geodesic Gauss map $\gamma$ projects this into the…

Let $M$ be an open Riemann surface and let $\Lambda\subset M$ be a closed discrete subset. In this paper, we prove the existence of complete conformal minimal immersions $M\to\mathbb{R}^n$, $n\ge 3$, with prescribed values on $\Lambda$ and…

We prove that if an RCD space has a regular isometric immersion in a Euclidean space, then the immersion is a locally bi-Lipschitz embedding map. This result leads us to prove that if a compact non-collapsed RCD space has an isometric…

Given an oriented immersed hypersurface in hyperbolic space $\mathbb{H}^{n+1}$, its Gauss map is defined with values in the space of oriented geodesics of $\mathbb{H}^{n+1}$, which is endowed with a natural para-K\"ahler structure. In this…

The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is…

In this paper we study isometric immersions $f:M^n \to {\mathbb {C}^{\prime}}\!P^n$ of an $n$-dimensional pseudo-Riemannian manifold $M^n$ into the $n$-dimensional para-complex projective space ${\mathbb {C}^{\prime}}\!P^n$. We study the…

We give a necessary and sufficient condition for a 2-dimensional Riemannian manifold to be locally isometrically immersed into a 3-dimensional homogeneous manifold with a 4-dimensional isometry group. The condition is expressed in terms of…